Multiphase Flow of Immiscible Fluids on Unstructured Moving Meshes

In this paper, we present a method for animating multiphase ﬂow of immiscible ﬂuids using unstructured moving meshes. Our underlying discretization is an unstructured tetrahedral mesh, the deformable simplicial complex (DSC), that moves with the ﬂow in a Lagrangian manner. Mesh optimization operations improve element quality and avoid element inversion. In the context of multiphase ﬂow, we guarantee that every element is occupied by a single ﬂuid and, consequently, the interface between ﬂuids is represented by a set of faces in the simplicial complex. This approach ensures that the underlying discretization matches the physics and avoids the additional book-keeping required in grid-based methods where multiple ﬂuids may occupy the same cell. Our Lagrangian approach naturally leads us to adopt a ﬁnite element approach to simulation, in contrast to the ﬁnite volume approaches adopted by a majority of ﬂuid simulation techniques that use tetrahedral meshes. We characterize ﬂuid simulation as an optimization problem allowing for full coupling of the pressure and velocity ﬁelds and the incorporation of a second-order surface energy. We introduce a preconditioner based on the diagonal Schur complement and solve our optimization on the GPU. We provide the results of parameter studies as well as a performance analysis of our method